In the classical balls-and-bins model, \(m\)  balls are independently and uniformly at random placed into \(n\)  bins one after another. In this paper, we are interested in the following variant, which is called the two-thinning model. For each ball, after a uniformly random bin, which is called the primary allocation, has been suggested, an overseer has the choice of either accepting this bin, or placing the ball into a new bin selected independently and uniformly at random, which is called the secondary allocation. In this model, the overseer is oblivious to the secondary allocation before deciding whether to accept the primary allocation. In contrast, in the well-known two-choice model, which was introduced in the seminal work [1]}, the overseer is aware of the secondary allocation and places the ball into the bin which contains fewer balls (break ties arbitrarily).
